Theoretical Background: Synthetic Aperture Radar Tomography of Forests
1 Introduction: Tomography and Three-Dimensional Forest Structure
Synthetic Aperture Radar (SAR) tomography, or TomoSAR, represents a powerful technique for reconstructing three-dimensional forest structure from radar observations. The term “tomography” derives from Greek: “tomos” (slice or section) and “graphia” (to describe or write). In essence, tomography enables the reconstruction of an object’s internal structure by analyzing its projections from multiple viewing angles.
The fundamental principle of tomography is well known from medical imaging, where techniques such as computed tomography (CT) or magnetic resonance imaging (MRI) reconstruct the internal structure of the human body by taking slice measurements from different angles. SAR tomography applies similar concepts to remote sensing of forests, using multiple radar acquisitions from different spatial positions to resolve the vertical structure within each image pixel.
As illustrated in Figure 1, projections taken from various angles contain information about the object’s internal structure. Through mathematical processing, including filtering techniques such as high-pass filters, a clear reconstruction of the original object can be achieved. While medical tomography scans a patient lying on a table through a rotating scanner, SAR tomography achieves similar results by acquiring radar images from different flight tracks or orbital positions.
2 Airborne and Spaceborne TomoSAR Systems
2.1 Airborne TomoSAR: High-Resolution Forest Mapping
Airborne SAR tomography systems provide the highest resolution observations of forest structure. The F-SAR system operated by the German Aerospace Center (DLR) exemplifies this capability, offering full polarimetric data in four wavelength bands (X-, C-, S-, and L-band).
Figure 2 demonstrates airborne TomoSAR’s ability to penetrate forest canopies and reveal both canopy structure and underlying terrain. When scanning flat terrain, a relatively uniform surface appears, while forest areas show distinct layering—returns from the canopy top, internal canopy structure, and the ground beneath. In particularly dense forests, complex vertical structure becomes visible, analogous to scanning a body in a medical tomography system.
The advantage of airborne TomoSAR is complete forest coverage with fine spatial detail. However, compared to airborne lidar systems, the vertical resolution of SAR tomography is typically coarser, though it provides unique information about electromagnetic scattering properties related to forest structure, moisture content, and biomass.
2.2 Spaceborne TomoSAR: The BIOMASS Mission
Until recently, no satellite mission was specifically designed for SAR tomography of forests. The European Space Agency’s (ESA) BIOMASS mission, launched in 2024, represents the first satellite dedicated to tomographic forest observations.
The BIOMASS satellite, depicted in Figure 3, operates at P-band (longer wavelength around 70 cm), which provides superior penetration through forest canopies compared to shorter wavelengths. The mission strategy involves repeated observations of the same location—approximately seven acquisitions within an 18-day period—enabling tomographic reconstruction of forest vertical structure.
A key innovation of P-band observations is the ability to image not only the upper canopy but also deeper structural elements including tree trunks and understory vegetation. This capability is essential for accurate forest height retrieval and above-ground biomass (AGB) estimation, particularly in dense tropical forests where shorter wavelengths may saturate.
3 Fundamental Principles: Phase and Interferometry
3.1 Phase as a Relative Distance Measure
The foundation of SAR tomography lies in interferometry, which exploits phase measurements to determine relative distances. In radar systems, phase represents the position within a wave cycle and can be understood as a relative distance measurement.
As illustrated in Figure 4, electromagnetic waves can be represented as rotating vectors (phasors) or sinusoidal functions. The key relationship is that one complete 2π rotation corresponds to one wavelength (λ) of propagation. If we know the phase difference (Δφ) between two observations, we can estimate the corresponding path difference:
\[\Delta\phi = \frac{2\pi}{\lambda} \cdot 2\Delta R\]
where ΔR is the range difference and the factor of 2 accounts for the two-way travel (to target and back).
Consider two radar sensors (B₁ and B₂) observing the same forest target. The phase difference between their observations encodes information about the relative distance from each sensor to the target. This principle forms the basis of SAR interferometry (InSAR).
3.2 SAR Interferometry (InSAR)
InSAR combines observations from two or more SAR acquisitions to measure surface topography and detect changes. The term “interferometry” refers to measuring properties through the interference of waves.
Figure 5 demonstrates the interferometric principle. When two wave sources emit coherent radiation, regions where the waves arrive in phase create maxima (constructive interference), while regions where they arrive out of phase create minima (destructive interference). This creates characteristic fringe patterns.
In spaceborne InSAR, even over completely flat terrain, fringe patterns emerge because different locations have different distances to the two sensor positions. Steep slopes facing the sensor produce dense fringes, while slopes facing away from the sensor show sparse fringes. These fringe patterns directly relate to surface topography.
3.3 From Phase Images to Elevation Models
While individual SAR phase images appear chaotic and random, the phase difference between two carefully selected acquisitions reveals meaningful patterns.
Figure 6 illustrates this transformation. A single SAR image’s phase appears as random noise, but when two images are properly aligned and their phases differenced, regular fringe patterns emerge (the wrapped interferogram). Through phase unwrapping and geometric processing, these fringes can be converted into topographic height information, producing a Digital Elevation Model (DEM).
3.4 Coherence: Quality of Interferometric Phase
Not all interferometric phase measurements are equally reliable. Coherence quantifies the quality and consistency of the interferometric phase.
Coherence (\(\hat{\gamma}\)) is calculated as:
\[\hat{\gamma} = \frac{\sum_{n=1}^{N} y_1 y_2^*}{\sqrt{\sum_{n=1}^{N} |y_1|^2 \sum_{n=1}^{N} |y_2|^2}}\]
where \(y_1\) and \(y_2\) are complex SAR observations, and the asterisk denotes complex conjugation. Coherence ranges from 0 (completely decorrelated) to 1 (perfectly correlated).
As shown in Figure 7:
- Coherence = 1.0: Perfect fringe visibility with minimal noise
- Coherence = 0.8: Clear fringes with some degradation
- Coherence = 0.4: Noisy fringes, information partially degraded
- Coherence = 0.2: Fringes barely visible, unreliable phase
High coherence indicates low phase noise and reliable distance measurements. In forest environments, coherence typically decreases over time as vegetation changes (temporal decorrelation) and varies with wavelength and environmental conditions. Coherence serves as a critical input parameter for TomoSAR processing, indicating which measurements are sufficiently reliable for tomographic inversion.
4 TomoSAR: Three-Dimensional Resolution Cells
4.1 From 2D Pixels to 3D Voxels
Conventional SAR imaging collapses all scatterers within a resolution cell into a single pixel. TomoSAR extends this to resolve the vertical distribution of scatterers within each pixel, effectively creating three-dimensional resolution cells.
Figure 8 illustrates the TomoSAR geometry. While conventional SAR creates a synthetic aperture along the flight direction (azimuth) to achieve fine azimuth resolution, TomoSAR adds multiple observations from different cross-track positions to create an additional synthetic aperture perpendicular to the line of sight. This elevation aperture enables vertical resolution.
The three-dimensional resolution cell is characterized by:
- Slant range resolution \(\delta_{sr} = \frac{c}{2W}\), where \(c\) is the speed of light and \(W\) is the signal bandwidth
- Azimuth resolution \(\delta_{az} = \frac{\lambda r_0}{2A_{az}}\), where \(r_0\) is the range distance and \(A_{az}\) is the synthetic aperture in azimuth
- Cross-range elevation resolution \(\delta_{cr} = \frac{\lambda r_0}{2A_{cr}}\), where \(A_{cr}\) is the synthetic aperture in elevation
- Vertical (height) resolution \(\delta_z = \delta_{cr} \sin(\theta_i)\), where \(\theta_i\) is the incidence angle
The tomographic resolution cell is oriented according to the radar viewing geometry, with the vertical extent depending on the number of flight tracks, their separation, and the wavelength used.
5 TomoSAR of Forests: Key Influencing Factors
The reflectivity profiles retrieved through TomoSAR depend on multiple interacting factors related to the sensor, platform, processing algorithms, and forest characteristics themselves.
As illustrated in Figure 9, TomoSAR provides vertical profiles of radar reflectivity, typically showing distinct peaks corresponding to the ground surface and canopy elements. However, the characteristics of these profiles are sensitive to:
- Sensor properties: Polarization and wavelength
- Platform configuration: Number and spacing of flight tracks or orbital passes
- Processing methods: Choice of inversion algorithm (e.g., Capon, beamforming)
- Forest characteristics: Vertical structure, biomass, and moisture content
5.1 Effect of Polarization
Different polarizations interact with forest structure in distinct ways. HH polarization (horizontal transmit and receive) tends to penetrate more deeply, showing stronger ground returns. HV or VH cross-polarization emphasizes volume scattering from branches and foliage. VV polarization shows intermediate behavior.
Figure 10 demonstrates these polarimetric differences. In the false-color composite on the left, different forest types and structures appear in different colors because each exhibits distinct polarimetric scattering properties. The vertical profiles on the right reveal:
- HH polarization: Strong ground peak, indicating penetration to the surface
- HV polarization: More distributed scattering throughout the canopy volume, with reduced ground contribution
- VV polarization: Intermediate characteristics between HH and HV
This polarimetric sensitivity enables scattering mechanism decomposition, separating ground scattering from volume scattering.
5.2 Scattering Mechanism Decomposition
Polarimetric TomoSAR can decompose the total radar return into contributions from different scattering mechanisms, primarily ground scattering and volume scattering.
As illustrated in Figure 11, advanced polarimetric-interferometric algorithms can separate:
- Ground scattering: Surface returns from the soil or understory ground, appearing as a concentrated peak at low elevation
- Volume scattering: Distributed returns from the canopy, appearing as a broader vertical distribution
This decomposition is particularly valuable for accurate ground surface estimation beneath forests, which is essential for biomass retrieval and change detection.
5.3 Effect of Wavelength
Different radar wavelengths penetrate forest canopies to different depths, resulting in distinct TomoSAR signatures.
